We present an extended linear quadratic regulator (LQR) design for continuous-time linear time invariant (LTI) systems in the presence of exogenous inputs with a novel feedback control structure. We first propose a model-based solution with cost minimization guarantees for states and inputs using dynamic programming (DP) that out-performs classical LQR with exogenous inputs. The control law consists of a combination of the optimal state feedback and an additional optimal term which is dependent on the exogenous inputs. The control gains for the two components are obtained by solving a set of matrix differential equations. We provide these solutions for both finite horizons and steady state cases. In the second part of the paper, we formulate a reinforcement learning (RL) based algorithm which does not need any model information except the input matrix, and can compute approximate steady-state extended LQR gains using measurements of the states, the control inputs, and the exogenous inputs. Both model-based and data-driven optimal control algorithms are tested with a numerical example under different exogenous inputs showcasing the effectiveness of the designs.

我们提出了一种扩展的线性二次调节器 (LQR) 设计，用于在存在具有新颖反馈控制结构的外源输入的情况下的连续时间线性时不变 (LTI) 系统。我们首先提出了一种基于模型的解决方案，使用动态规划 (DP) 为状态和输入提供成本最小化保证，该解决方案优于具有外生输入的经典 LQR。控制律由最优状态反馈和附加最优项取决于外源输入。这两个分量的控制增益是通过求解一组矩阵微分方程获得的。我们为有限范围和稳态情况提供这些解决方案。在论文的第二部分，我们制定了一种基于强化学习 (RL) 的算法，该算法除了输入矩阵之外不需要任何模型信息，并且可以使用状态、控制输入、和外源输入。基于模型和数据驱动的最优控制算法都在不同的外源输入下用数值例子进行了测试，展示了设计的有效性。